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# Chapter 3 SMITH CHART Outlines - PDF by s42gs6

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```									               Chapter 3
SMITH CHART

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Outlines
Construction of Smith Chart
Operations and Applications of Smith Chart

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SMITH CHART:

By P. H. Smith in 1939

A graphical tool for the calculation of impedance
and reflection coefficient

A useful tool of visualizing the concepts and
procedures of RF circuit designs

http://www.sss-mag.com/smith.html

3

Construction of Smith Chart

lossless TL : Z 0 : real,   γ = jβ
Γin             Z0 , β          ΓL       ZL
Z − Z0   Z −1
Γ =          =
Z in                      Z in = Z L               Z + Z0   Z +1

let         Γ = Γ r + jΓ i ,    Z = Z / Z 0 = r + jx

2                2
        r            1 
equation for resistance r :  Γ r −       + Γi = 
2

       1+ r         1+ r 

2
      1   1
(Γ r   − 1) +  Γ i −  = 2
2
equation for reactance x :
      x  x

4
for passive load : | Γ |≤ 1,    r :0 ~ ∞,     x : −∞ ~ 0 ~ ∞

| Γ |≤ 1         Γ within the unit circle
£F   i
centered at the origin
(0, 1)

2                 2
       r            1 
 Γr −       + Γi = 
2

      1+ r         1+ r                     x = 0.3               x=1

x=2
resistance circles:
1                                                             (1, 0)
center ( r , 0) , radius =
1+ r               1+ r          x=0                                               £F
r=2                    r

2                                               r=1
      1   1
(Γ r   − 1) +  Γ i −  = 2
2

      x  x                                           r = 0.5
r=0         x = -1
reactance circles:
1                                                 £F       =1
center (1, 1 ) , radius =                                                         r
x              |x|                          Γ plane

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Smith Chart

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Important Notes on Smith Chart
r = 0, pure reactance
r=1

r < 1 : R < Z0

x > 0 : inductive
Γ = −1, SC                                     r > 1 : R > Z0             Γ = 1, OC

x > 0 : capacitive                                Γ = 0, matched
x = 0, pure resistance

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Basic Information by Smith Chart
P                                               x

φ
Γ          Z0            ZL
P

Z
r

Γ =| Γ | e jφ
Z = Z0 ( r + jx)

|Γ|,
VSWR,
RL

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Y Z0 1 1 − Γ
y=     = = =      = g + jb
Y0 Z z 1 + Γ

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ZY Smith Chart

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1.0                                 by Z Smith Chart :
0.5                       zP = 1 + j
yP = 0.5 − j0.5

P
by Y Smith Chart :
yP = 0.5 − j0.5
0.5         1.0       0.5

0.5

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Finding z, y, Γ Along Transmission Line
d
Q                  P                    ar tor
tow nera
ge
ΓQ                 ΓP          ZL
k
P

ZQ                 ZP                                                              = kλ

= kλ                                                           Q

ΓQ = Γ Pe− j2β
to
− j 4 kπ               lo wa

Rotating 360 degree ⇒ = 0.5λ
Quarter wavelength transformer:              = λ /4   zQ = 1/ zP = yP

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Example

Determine the length of a 50 Ohm open-circuited transmission
line shown in figure so that the input impedance is Z IN (l ) = j100Ω

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Series Element

series connect

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Shunt Element

shunt connect

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Example

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Example

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Example

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Example

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Practice :

A 50Ω lossless transmission line is terminated by a
100 Ω resistor in series with a 2 nH inductor. The
electrical length of the line is 270° at 2.4 GHz. Plot Zin
and Γin verse frequency in Smith chart for f = 1.2 to 3.6
GHz by ADS. Find the Zin and Γin of the RL-terminated
transmission line at f = 3.2GHz.

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